1. Field
The subject disclosure relates to apparatus and methods for measuring the permeability and/or the porosity of a solid sample. The subject disclosure more particularly relates to apparatus and methods for measuring the permeability and/or porosity of a rock sample having an “ultra-low” permeability (in the range of hundreds of nanoDarcies to 100 milliDarcies) obtained from a geological formation, although it is not limited thereto.
2. State of the Art
For creeping incompressible fluid flow, with Reynolds number much less than unity, the Stokes equation in terms of microscopic velocity vector u is∂iP=ρgi+∂j{μ(∂jui+∂iuj)},  (1)where the subscripts i and j are cartesian component indices, ρ is the density, P is the pressure, g is the gravitational acceleration, and μ is the shear or dynamic viscosity. For steady creeping flow in porous media, these equations scale up to macroscopic velocity components
                                          v            i                    =                                    -                              k                μ                                      ⁢                          (                                                                    ∂                    i                                    ⁢                  P                                -                                  ρ                  ⁢                                                                          ⁢                                      g                    i                                                              )                                      ,                            (        2        )            where the fluid pressure P refers to a local volume average quantity in the Darcy formulation. This Darcy scaling is valid for a large range of pore sizes and velocities. As a consequence of the Stokes equation, the derivation of Darcy's law in the continuum form is shown to be valid through homogenization or local volume averaging.
It can be taken for granted that the validity of Darcy's law extends to slowly varying unsteady compressible flow. Accordingly, νi and P may vary with time, and for isothermal conditions ρ may be evaluated from an explicit functional dependence on P. This approach, while prevalent, is often used without stating restrictions. For the approach to be valid, it is necessary that on the scale on which permeability is defined, variation in density should be negligible. The time-scale for external variation in fluxes or pressure should also be very large compared to the time scale for local establishment of Darcy's law.
Starting with Darcy's gravity-head based determination of proportionality between flux and pressure drop, experiments to determine permeability are well documented. It is assumed that for inert media, k is independent of the fluid, μ being the normalizing factor containing the fluid property. Therefore, gas and liquid permeabilities are expected to be the same, unless the mean free path is comparable to pore size.
For gas permeability, a series of steady-state experiments relating flow-rate and pressure difference across the length L of the rock may be derived by combining continuity, equation of state, and Darcy's law. For an ideal gas, the result for isothermal flow is
                              q          s                =                  k          ⁢                                                    (                                                      P                    L                    2                                    -                                      P                    R                    2                                                  )                            ⁢                              AT                s                                                    2              ⁢                                                          ⁢                              TP                s                            ⁢              μ              ⁢                                                          ⁢              L                                                          (        3        )            where PL and PR are respectively the pressures on the “left” or upstream, and “right” or downstream sides of the rock, q is the flow rate, T is the temperature, and L is the length of the rock sample, A is the cross sectional area of the rock sample, with the subscript s referring to standard conditions. A series of appropriately chosen flow rates to stay within the optimal regime and the resulting steady-state pressure measurements allows for a determination of the permeability in a best-fit sense fairly accurately (see, e.g., U.S. Pat. No. 5,832,409 to Ramakrishnan et al.) The accuracy estimate using the steady-state method is better than 1 percent.
Two problems preclude the steady-state permeability measurement for rocks below about one mD. First, imposition of controllable or measurable flow rates results in excessive pressure drop, resulting in unknown nonlinear corrections. Second, the establishment of successive steady-states becomes onerously long, rendering the experiment impractical.
Analogous to well-testing pressure transient methods (see, e.g., Raghavan, R., Well Test Analysis, Prentice Hall, N.J. (1993)), but in the laboratory, the transient build-up of pressure for an imposed flow-rate may be studied and the permeability inferred relatively quickly. Such methods are again error-prone because the build-up is affected by the line volumes, and the magnitude of the build-up needs to be determined a priori in order to reduce the nonlinear effects. It is the nonlinearity of both the rock behavior and the gas that precludes obtaining accurate transport properties with a step-rate method. For very low permeability rocks, the rates are too small to be measured reliably.
A concept for a system for measuring permeability of granites has been described by Grace et al., “Permeability of granite under high pressure,” J. Geophysical Res., 73 (6) 2225-2236 (1968). However, the analyses of Grace et al. do not take into account dead-volumes of the hardware in which the core is located, and the nonideality of the gas saturating the medium, and also do not provide a complete mathematical solution to the transient problem. As a result, the analyses of Grace et al. do not permit for accurate results.
Based on the methods of Grace et al., Hsieh et al., proposed a solution in terms of Laplace and inverse Laplace transforms. See, Hsieh et al., “A transient labaoratory method for determining the hydraulic properties of ‘tight’ rocks,” Int. J. Rock Mech. Min. Sci. and Geomech, 18, 245-252, 253-258 (1981). The analysis of Hsieh et al., however, does not include nonideality of the gas explicitly and its influence of the transient characteristics. Dead-volume connected to the core is also not included. Interpretation is based on what Hsieh et al. call as an early-time semi-infinite solution or the late-time single exponential result. Hsieh et al. analyze upstream and downstream pressures or their difference with respect to the final pressure. As a result, the solution of Hsieh et al., does not permit for accurate results.